Estimation of Ideal Points using Cast Vote Records

Introduction

How representative is a government of its people?

Cast Vote Records

Distribution of Cast Vote Records

Methodology

Continuous, latent outcome with categorical indicators –> IRT model

\[ \pi_{j,k(c)} = logit^{-1} (\gamma_{k(c)} (\alpha_j - \beta_{k(c)})) \]

where \(j\) indexes individuals and \(k(c)\) indexes a specific candidate \(c\) in a race \(k\).

• \(\gamma\) is the “discrimination” parameter

  • \(=1\) in the “Rasch” model, implies all races inform \(\alpha\) equally

• \(\alpha\) is the latent trait

• \(\beta\) is the “difficulty” parameter

• \(\kappa = \frac{\alpha}{\beta}\) is the cutpoint between each choice

Modeling \(\pi\)

• \(\pi_{i, j(c)}\) as a Binary Variable

  • \(Pr(\pi_{i, j(c)} = 1)\) , where \(=1\) implies the voter chose the Republican candidate

    • Arbitrary choice of Republican, but makes scale normal

• \(\pi_{i, j(c)}\) as a Categorical Variable

  • \(Pr(\pi_{i, j(c)} = c)\) for each candidate

  • Now I also estimate discrimination and difficulty parameters by candidate

    • Everything is now relative to the baseline, makes interpretation confusing

  • Subset to just Adams County, Colorado

Identification and Priors

• The scale and location of \(\alpha\) are unidentified (can add or multiply a constant with no change in likelihood)

  • Set prior to \(\mathcal{N}(0, 1)\)

• \(\gamma\)’s sign can also flip around with relation to the sign of \(\alpha\).

  • Constrain \(\gamma > 0\)

• Priors

  • Uninformative, standard priors for other parameters

  • Future work: Informative priors using party or NOMINATE/DIME scores to help with identification and speed convergence

brms Binomial Models

form_rasch <- bf(
  choice_rep ~ 1 + (1 | race) + (1 | cvr_id),
  family = brmsfamily("bernoulli", link = "logit")
)

form_2pl <- bf(
  choice_rep ~ beta + exp(loggamma) * alpha,
  nl = TRUE,
  alpha ~ 0 + (1 | cvr_id),
  beta ~ 1 + (1 |i| race),
  loggamma ~ 1 + (1 |i| race),
  family = brmsfamily("bernoulli", link = "logit")
)

Stan Categorical Models

// Priors
mu_beta ~ cauchy(0, 5);
alpha ~ std_normal();

for (k in 1:K){
  beta_raw[k,] ~ normal(0, sigma_beta);
  gamma_raw[k,] ~ lognormal(0, sigma_gamma);
}

sigma_beta ~ cauchy(0, 5);
sigma_gamma ~ cauchy(0, 5);

// Likelihood
for (j in 1:J) {
  matrix[K, C] logits = rep_matrix(-1e8, K, C);
  for (k in 1:K) {
    if (eligibility[j, k] == 1) {
      for (c in 1:C) {
        if (candidates[k, c] == 1) {
          logits[k, c] = gamma[k, c] * alpha[j] - (beta[k, c] + mu_beta);
        }
      }
      target += categorical_logit_lpmf(votes[j, k] | logits[k]');
    }
  }
}

Convergence

Binary Results - \(\alpha\)

Binary Results - \(\beta\) and \(\gamma\)

Categorical Results – \(\alpha\)

Categorical Results – \(\alpha\)

Categorical Results - \(\gamma\)

Categorical Results - \(\beta\)

Future Work

• Informative priors from DIME or NOMINATE scores

• Multiple dimensions

• Connect this better to a question about representation